
in which case it is proved in [1]} that \(d_{[]}\)  is a cone distance.
The case \(\phi =\)  is equivalent to partial unbalanced OT, which produces discontinuities (because of the non-smoothness of the divergence) between regions of the supports which are being transported and regions where mass is being destroyed/created.
Note that [2]} does not mention that such cone cost \(d_{[]}\)  defines a distance, although it can be proved without a conic lifting that partial OT defines a distance as explained in [1]}.
